It's actually possible to extend complex numbers to handle 3D rotations and translations. The 3D analog of the complex numbers are well known as the quaternions, but there also exist the dual-quaternions which are capable of describing any proper rigid transformation, ie rotation and translation. There's also an interesting way to extend these to higher dimensions as well as other types of transformations. While the components grow faster than matrices, doubling with each additional underlying dimension rather than going to the next square, they provide much smoother interpolation. I actually noticed a few times in this video where an object appeared to shrink as it was moving before ending up at the same size as it started.
Indeed, and interestingly, the complex numbers are the even sub-algebra of the 2D geometric algebra and the quaternions are the even sub-algebra of the 3D geometric algebra.
@@cstockman3461 Every geometric algebra is the even subalgebra of a higher geometric algebra. The dual-quaternions are the even subalgebra of 3D projective geometric algebra, and 3D vanilla geometric algebra is the even subalgebra of spacetime algebra. Geometrically, the PGA interpretation can apply to other algebras, in which case not only in VGA algebraically a subalgebra of PGA, but geometrically too, with a given VGA multivector representing the exact same subspace and transformation in PGA. Geometric algebra isn't really an algebra. It's more like a Matryoshka Doll of algebras. Even the basic Real numbers can be considered the even subalgebra of the complex numbers. Geometric algebras all the way down... And going all the way _up_ eventually brings you to the mythical Universal Geometric Algebra (UGA) aka Cl(∞,∞)
Very cool. Now, it is just a small step to quaternions😀. By the way, since there was a short blender clip inside the video, I just wanted to mention that I'm working on a library that realizes much of the manim tools inside blender. If you are interested, let me know.
so everyone knows, nowdays is common to hear that matrices do transformations, which is misleading what is actually happening is that in algebra, there is a concept called linear transformations that are just equations with some constraints this equations end up as a system of equations with each equation having a series of products between constants and variables, such as: a*x + b*y = k c*x + d*y = h and all linear transformations have a matrix representation, which, in this case, is: | a b | | x | | k | | c d | * | y | = | h | so the matrix abcd represents a specific linear transformation over some coordinates xy this transformation can be whatever you want, but if you want specific properties for this transformations, you can specify it in the original equations, figure them out and then the matrix comes in free
If you wonder why your viewing stats tell you that you lose people around 3:20, it’s because you start using the expression “i-hat” etc. without any prior definition. Especially for foreign viewers, this “hat” qualifier sounds weird and childish. Yes I know it stems from the caret notation being read as 🎩. Vector i is the proper way to say. To add to the confusion, you failed to label the î. Being a good teacher is mostly about not forgetting the time you were like your students (and weren’t comfortable with all the conventions)
The scaling factor being a higher dimensional shear operation seems so obvious in retrospect.
Thank you so much for the animation!
Great graphics and explanation. I thought I was watching a 3 Blue 1Brown video at times. Well done!
any translation in low dimensions can be represented as a transformation in higher dimensions (n+1). Great illustration !
Great video!
One off the best I've seen on the topic!
@@TheRickCh Thank you!
It's actually possible to extend complex numbers to handle 3D rotations and translations. The 3D analog of the complex numbers are well known as the quaternions, but there also exist the dual-quaternions which are capable of describing any proper rigid transformation, ie rotation and translation. There's also an interesting way to extend these to higher dimensions as well as other types of transformations. While the components grow faster than matrices, doubling with each additional underlying dimension rather than going to the next square, they provide much smoother interpolation. I actually noticed a few times in this video where an object appeared to shrink as it was moving before ending up at the same size as it started.
Indeed, and interestingly, the complex numbers are the even sub-algebra of the 2D geometric algebra and the quaternions are the even sub-algebra of the 3D geometric algebra.
@@cstockman3461 Every geometric algebra is the even subalgebra of a higher geometric algebra. The dual-quaternions are the even subalgebra of 3D projective geometric algebra, and 3D vanilla geometric algebra is the even subalgebra of spacetime algebra.
Geometrically, the PGA interpretation can apply to other algebras, in which case not only in VGA algebraically a subalgebra of PGA, but geometrically too, with a given VGA multivector representing the exact same subspace and transformation in PGA.
Geometric algebra isn't really an algebra. It's more like a Matryoshka Doll of algebras. Even the basic Real numbers can be considered the even subalgebra of the complex numbers. Geometric algebras all the way down... And going all the way _up_ eventually brings you to the mythical Universal Geometric Algebra (UGA) aka Cl(∞,∞)
Super cool video, really helpful to build intuition.
Fan of your channel!
when a student tries to become a teacher. thats you and this video. NICE
So cool! It's definitely going to be helpful for me
Great video! I didn't knew homogeneous coordinates intuitively. ! Nice visuals
Thank you so much!
Great video
Linear algebra is still a very new concept for me but this video was very nifty! Awesome work :)
awesome job!
excellent explanations
Wow😲. So helpful to me. Thanks a lot.
Great explanation, thanks a lot!!!
this is exactly what i wanted!!
Now make a 4D game using 5D matrices (5x5 matrices)
thats EZ
CodeParade be like
Thank you!
Really nice
Very cool. Now, it is just a small step to quaternions😀. By the way, since there was a short blender clip inside the video, I just wanted to mention that I'm working on a library that realizes much of the manim tools inside blender. If you are interested, let me know.
excellent
It is possible to visualize 4D geometry, and even to show it graphically and animate it.
so everyone knows, nowdays is common to hear that matrices do transformations, which is misleading
what is actually happening is that in algebra, there is a concept called linear transformations that are just equations with some constraints
this equations end up as a system of equations with each equation having a series of products between constants and variables, such as:
a*x + b*y = k
c*x + d*y = h
and all linear transformations have a matrix representation, which, in this case, is:
| a b | | x | | k |
| c d | * | y | = | h |
so the matrix abcd represents a specific linear transformation over some coordinates xy
this transformation can be whatever you want, but if you want specific properties for this transformations, you can specify it in the original equations, figure them out and then the matrix comes in free
If you wonder why your viewing stats tell you that you lose people around 3:20, it’s because you start using the expression “i-hat” etc. without any prior definition. Especially for foreign viewers, this “hat” qualifier sounds weird and childish. Yes I know it stems from the caret notation being read as 🎩. Vector i is the proper way to say. To add to the confusion, you failed to label the î. Being a good teacher is mostly about not forgetting the time you were like your students (and weren’t comfortable with all the conventions)
great video! at 6:32 please use 'dots' instead of 'x' for matrix multiplication :)
why is that
@@bbrother92 X implies the cross product which is a different type of multiplication
@@matthewjames7513 im p sure cross product is only defined for vectors